Every ratio has a reciprocal, found by flipping the fraction. Cosecant is 1 ÷ sine, secant is 1 ÷ cosine, cotangent is 1 ÷ tangent. The prefixes cross over on purpose.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
In a right triangle the leg opposite θ is 6 cm and the leg adjacent to θ is 8 cm. Using only Pythagoras and the ratios from the last lesson, what is hypotenuse ÷ opposite, as a decimal to two places?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
The three names
csc θ = hypotenuse ÷ opposite. sec θ = hypotenuse ÷ adjacent. cot θ = adjacent ÷ opposite. Each is the original ratio upside down.
The pairing looks wrong
Secant pairs with cosine, and cosecant pairs with sine. Read the third letter: co-sec-ant has an s, so it belongs to sine; se-c-ant has a c, so it belongs to cosine.
Evaluating all six
Legs 3 and 4, hypotenuse 5, angle opposite the 3: sin = 3/5, cos = 4/5, tan = 3/4. Flip each: csc = 5/3, sec = 5/4, cot = 4/3. Find the first three, then flip.
Sizes
Sine and cosine never exceed 1, so cosecant and secant are never below 1. Tangent can be anything, so cotangent can too. A secant of 0.6 is impossible.
Where they break
A reciprocal is undefined wherever the original ratio is zero. sin 0° = 0, so csc 0° does not exist. cos 90° = 0, so sec 90° does not exist.
Why bother
They shorten formulas. The derivative of tangent is sec² x, far tidier than 1 ÷ cos² x every time. The identities tan² θ + 1 = sec² θ and 1 + cot² θ = csc² θ are written with them.
Step 2: Try It Yourself
Tap and try it out.
Opposite 5, adjacent 12. Find all six ratios of the angle
- 1hypotenuse = √(25 + 144) = 13Pythagoras
Step 0 of 3
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: In Real Life
Old navigation tables
Navigation and optics tables list the secant, 1/cos, because dividing by a cosine was slow by hand. Reciprocal ratios are the same triangle, read the other way up.
Step 4: Watch an Example
One step at a time.
Watch Noor Find All Three Reciprocals
In a right triangle the leg opposite θ is 7 cm and the hypotenuse is 25 cm. Noor wants csc θ, sec θ and cot θ.
- Step 1
The two sides given are the opposite, 7, and the hypotenuse, 25. The cosecant could be written from those two alone, but the secant and the cotangent both want the adjacent leg, which is missing, so Pythagoras supplies it: √(625 − 49) = √576 = 24 cm. With the third side down, all six ratios are available.
Watch Dev Start From a Reciprocal
Dev is told only that cot θ = 8/15 for an acute angle θ, and is asked for csc θ and sec θ. He writes the first line; you supply the next three.
- Step 1
Cotangent is adjacent over opposite, so cot θ = 8/15 says the leg adjacent to θ is 8 and the leg opposite it is 15. Dev takes those as centimetres. Any triangle with legs in that proportion has this same angle, and so has these same six ratios.
Step 5: Your Turn
Practice makes it stick.
The Ramp Quote
Problem 1 of 2
A ramp rises 3 feet from the ground to a doorway. For its slope angle θ, csc θ = 8.5. The sloping board is the hypotenuse, so its length is the rise multiplied by the cosecant. Timber costs $12 a foot. What does the board cost, in dollars?
dollars
The Rafter Order
Problem 2 of 2
A rafter is the sloping timber of a roof. Each rafter covers a horizontal run of 12 feet, and for the roof's pitch angle θ, sec θ = 1.25. The run is the adjacent side and the rafter is the hypotenuse. Rafter timber costs $9 a foot and the roof needs 20 rafters. What does the timber cost, in dollars?
dollars
Flip the Ratio
Write the basic ratio first, then turn it over. The working is shown once your answer is in.
1 of 5
sin θ = 0.8. What is csc θ, as a decimal?
2 of 5
For angle θ the opposite leg is 9 cm and the hypotenuse is 41 cm. What is csc θ, as a decimal to two places?
3 of 5
For angle θ the adjacent leg is 20 cm and the opposite leg is 21 cm. What is cot θ, as a decimal to two places?
4 of 5
sec θ = 2.5. What is cos θ, as a decimal?
5 of 5
cot θ = 1.6. What is tan θ, as a decimal?
Whose Step?
One line in each piece of working is a slip. Find it, then put it right.
1 of 6
Priya was asked for sec θ in a triangle whose opposite leg is 24 cm and adjacent leg is 7 cm. She finished with 25/24. One line is a slip. Which?
Priya's working: opposite 24, adjacent 7 — find sec θ
- 1hypotenuse = √(576 + 49) = √625 = 25
- 2sin θ = 24/25, cos θ = 7/25
- 3sec θ = 1 ÷ sin θ
- Answersec θ = 25/24
2 of 6
Put Priya's secant right. What is sec θ, as a decimal to two places?
3 of 6
Dev was asked for cot θ when tan θ = 0.5, and answered 26.57. One line is a slip. Which?
Dev's working: tan θ = 0.5 — find cot θ
- 1tan θ = 0.5
- 2θ = tan⁻¹ 0.5 = 26.57°
- Answercot θ = 26.57
4 of 6
What is cot θ when tan θ = 0.5?
5 of 6
Ana was asked for csc θ and sec θ in a triangle whose opposite leg is 12 cm and hypotenuse is 13 cm. One line is a slip. Which?
Ana's working: opposite 12, hypotenuse 13 — find csc θ and sec θ
- 1adjacent = √(169 − 144) = √25 = 5
- 2sin θ = 12/13, cos θ = 5/13
- 3csc θ = 13/12, sec θ = 5/13
- Answercsc θ ≈ 1.08, sec θ ≈ 0.38
6 of 6
Put Ana's secant right. What is sec θ, as a decimal?
Pick the Move
Choose the first move before working anything out.
1 of 2
A question gives cos θ = 0.32 and asks for sec θ. Which first move gets there?
2 of 2
For angle θ the opposite leg is 12 cm and the adjacent leg is 35 cm, and the question asks for csc θ. Which first move gets there?
Same Value?
Judge these without working out a single number.
1 of 2
Select every expression that has the same value as csc θ.
2 of 2
θ is an acute angle in a right triangle. Select every statement that is true of it.
Step 6: Quick Check
Show what you know.
Question 1 of 2
Which ratio is the reciprocal of cosine?
Question 2 of 2
cos θ = 0.5. What is sec θ?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Find the acute angle θ for which cot θ and tan θ are the same number. What is θ, in degrees?
degrees
Stretch 2 of 2
An acute angle has sec θ = 2 exactly. That one fact fixes every other ratio of the angle. What is csc θ, to two decimal places?
What You Learned
- Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent.
- The prefixes cross over: secant pairs with cosine.
- Find the three basic ratios, then flip each.
- A reciprocal is undefined wherever the original ratio is zero, and never below 1 for csc and sec.
For the grown-up
The misconception this lesson targets is that the reciprocal and the inverse are the same thing, because both are written with a −1. sin⁻¹ x is the angle whose sine is x; (sin x)⁻¹ is 1 ÷ sin x; the two are almost never equal, and a student who blurs them will answer a ratio question with an angle in degrees. Its cousin is the pairing, since cosecant sounds like a partner for cosine and is matched with it about as often as not. Two habits close both gaps: read the third letter of the name — co-sec-ant has an s for sine, se-c-ant a c for cosine — and check the size, because for an acute angle a cosecant or a secant below 1 is impossible.